Elsevier

Cement and Concrete Research

Volume 79, January 2016, Pages 353-365
Cement and Concrete Research

A new model for the cracking process and tensile ductility of Strain Hardening Cementitious Composites (SHCC)

https://doi.org/10.1016/j.cemconres.2015.10.009Get rights and content

Abstract

Strain Hardening Cementitious Composites (SHCC) are materials exhibiting tensile hardening behavior up to several percent strain accompanied by the formation of fine multiple cracks. Their tensile ductility is governed by the spacing and opening of cracks, which depend on the stress transfer between the fibers and the matrix. In this article, a new analytic model which takes into consideration the effects of non-uniform matrix strength, post-cracking increase in fiber bridging stress and fiber rupture on stress transfer and multiple cracking behavior of SHCC is developed. Using material parameters within the range reported in the literature, simulation results can reach reasonable agreement with test data on SHCC for two different fiber contents. The effect of fiber length on tensile behavior of SHCC is then simulated to illustrate the applicability of the model to material design. The new model should be helpful to the micromechanics-based design of SHCC for various ductility requirements.

Introduction

The brittleness of cementitious materials is their major limitation. Even with steel reinforcements, the poor resistance to cracking makes cementitious members vulnerable to seismic loading and impact. In addition, the formation and opening of cracks facilitate the penetration of water and chemicals which greatly affects the durability of the structure. To control cracking, fibers can be added. With increasing fiber content, the cementitious composite transforms from a quasi-brittle material, with tension-softening behavior after a crack is formed, into a Strain Hardening Cementitious Composite (SHCC), with hardening behavior up to several percent strain, accompanied by the formation of multiple cracks with tightly controlled openings [1]. With proper design guided by micromechanics, SHCC can be made with moderate fiber content of 2% or less in total volume of materials [1].

The high ductility of SHCC is achieved by the formation of multiple cracks which is possible if the fibers can carry a higher tensile force than that corresponding to the first cracking of the cementitious matrix. After the first crack is formed, the applied tension can continue to increase. At the crack, the matrix stress drops to zero, and the stress in the bridging fibers has to increase to maintain equilibrium. Away from the crack plane, the additional stress taken by fibers will be transferred back to the surrounding matrix through the fiber/matrix interface. The stress in the matrix increases with distance from the crack, and at a critical transfer distance (xd), the stress reaches the matrix strength again. At any distance larger than xd from a certain crack, another crack can form. This is the mechanism for multiple cracking to take place. After all the cracks are formed, the minimum crack spacing is xd while the maximum spacing is 2xd [2].

For the hardening composite, the ultimate tensile strength is reached when the fiber bridging stress reaches its maximum value. The corresponding ultimate tensile strain is governed by the crack spacing. Theories for crack spacing calculation in fiber reinforced brittle matrix composites have been established in the 1970s [2], [3], [4] and further developed for SHCC in the 80s [5], [6] and 90s [7], [8]. In all existing theories, the cracking strength of the composite is assumed to be uniform along the whole member. Under this assumption, once the cracking strength of composites is reached, all cracks will form simultaneously and the value of xd is the same for each crack. This is inconsistent with experimental observation where multiple cracks form in sequence with increasing load. In reality, the cracking strength at different cross-sections along a member varies, and the plausible reasons include: 1) variation of flaw size in each cross-sectional plane, 2) variation of fiber volume fraction in each plane, and 3) interaction between matrix cracks [9]. As xd is the distance from a crack for the matrix strength to be reached again, when cracks form sequentially with increasing load, the value of xd also varies. To properly consider this effect in the calculation of crack spacing, xd can be derived in terms of the crack opening displacement (COD) which has a one-to-one relation to the crack bridging stress in the rising part of the stress vs COD curve of the fiber composite.

In practice, although fiber usually has higher tensile strength than cementitious matrix, fiber rupture often occurs at high tensile stress after matrix cracks [10], [11], [12]. The rupturing of fibers clearly has an effect on the crack bridging stress as well as xd, because stress transfer becomes less effective with a smaller number of load carrying fibers. However, this was not considered in existing theories for crack spacing.

In this paper, with the assumption of non-uniform matrix first-cracking strength along the member, a new model to predict the crack formation (with non-uniform spacing) in SHCC is proposed. Knowing the number and opening of the cracks, the stress–strain relation is also derived. In the model, the combined effects of increasing COD and fiber rupture on xd are considered. The simulated stress–strain relations will be compared with experimental results. Also, numerical simulation is carried out with the model to study the effect of fiber length on stress–strain behavior. The ultimate goal of this study is to establish a framework to guide the design of SHCC with specified tensile ductility through the proper simulation of multiple cracking.

Section snippets

Existing theories on transfer distance

As the new model to be developed is an extension of existing theories, a brief review of existing work is carried out first. The tensile ductility of a composite is dependent on the crack spacing. At saturated cracking (i.e., when the number of cracks stops increasing), the final crack spacing is theoretically between xd and 2xd. Based on the analogy with minimum average spacing between cars of length xd and parked randomly along an infinite line, the average crack spacing was derived as 1.337xd

Effects of crack opening and fiber rupture

The previous section reviews the calculation of transfer distance xd in existing theories. In this section, the following limitations are addressed:

  • 1)

    As the strength of matrix is not uniform along the member, a method to consider the strength variation needs to be developed. The variation of σmuVm in Eq. (2) will affect the transfer length xd.

  • 2)

    With strain hardening behavior, the crack bridging stress can increase significantly beyond the first cracking strength. With increasing bridging stress,

Simulation of uniaxial tensile test with non-uniform strength

In this section, the tensile behavior of SHCC with material parameters listed in Table 1 is determined by simulating the stress vs strain relation in a uniaxial tensile test. The simulation approach will be described first, followed by comparison with experimental data. It should be highlighted that the purpose of the comparison is not to prove that the model is perfect, as some of the model parameters (from Table 1) are not directly measured for the SHCC employed in the experiments. Our

Parametric study on the effect of fiber length

To further demonstrate the applicability of the model, a parametric simulation is carried out to evaluate the effect of PVA fiber length on the tensile performance. From the manufacturing point of view, the fiber length is a parameter that is easy to change in practice. Therefore, it will be helpful if we can find an optimized length from the simulation.

The bridging stress vs crack opening curves for different fiber lengths (10 mm, 12 mm and 14 mm) with the same fiber volume fraction of 2%, are

Conclusions

In this study, based on the mechanism of stress transfer, a new model to predict the cracking process and stress–strain relation of SHCC is developed. The model takes into consideration the strength variation along the member, the increase of crack bridging stress in the hardening regime as well as the possibility of fiber rupture. Stress–strain relation is simulated with increasing applied stress starting from the first cracking strength. By using material parameters within the range reported

Acknowledgment

Financial support of this work by the Hong Kong Research Grant Council through GRF 615411 is gratefully acknowledged.

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